浙江省专升本考试-函数、极限与连续性
2022-04-18 19:54:32 0 举报AI智能生成
浙江省专升本考试-函数、极限与连续性
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大纲/内容
<font color="#000000">求定义域</font>
<font color="#000000">根据函数式确定定义域所满足的不等式，然后直接通过不等式解出x的范围或根据图像求出x的范围</font>
<font color="#000000">题型一：具体函数</font>
1.分式分母不为零
2.偶次根下大于等于零
3.对数的真数大于零
4.任何非零实数的0次幂都为1
5.三角函数tanx,cot的定义域
6.反三角函数arcsinx，arccosx的定义域
题型二：抽象函数
方法总结：同等地位替换
1.已知f(x)的定义域，求f[g(x)]的定义域
2.已知f[g(x)]的定义域，求f(x)的定义域
注
<font color="#000000">1.不要化简函数</font>
2.定义域结果表示成区间或集合的形式
<font color="#000000">相同函数判断</font>
定义域（不化简）
对应法则（先化简）
函数的表达形式
1.显函数：y能由x直接表示出来
2.隐函数：y不能由x直接表示出来，而是由方程f(x,y)=0表示，考点：求导
3.分段函数：将定义域分成几部分表示一个函数,<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=|x|=\begin{cases}-x,x<0 \\0,x=0\\x,x>0\end{cases}"><span></span><span></span></span>,考点：分段点处的极限、连续性、可导性
4.参数函数：y和x没有直接关系，而是由共同的参数表示，<span class="equation-text" data-index="0" data-equation="x^2+y^2=1" contenteditable="false"><span></span><span></span></span> <span class="equation-text" contenteditable="false" data-index="1" data-equation="\Leftrightarrow"><span></span><span></span></span> <span class="equation-text" contenteditable="false" data-index="2" data-equation="\begin{cases}x=cost \\y=sint \end{cases}"><span></span><span></span></span>,考点：求导
注
显函数是隐函数的一种特例，将隐函数变成显函数的过程叫隐函数的显化
几个特殊函数
1.符号函数:<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=sgnx=\begin{cases}1,x>0 \\0,x=0 \\-1,x<0\end{cases}"><span></span><span></span></span>
对于任意实数x,都有关系成立:<span class="equation-text" contenteditable="false" data-index="0" data-equation="x=sgnx\cdot|x|"><span></span><span></span></span>
2.取整函数:<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=[x]"><span></span><span></span></span><br>
[x]表示不超过x的最大整数
范围：<span class="equation-text" contenteditable="false" data-index="0" data-equation="x-1\leq[x]\leq x"><span></span><span></span></span>
方法：向左取整
3.狄利克雷函数:<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=D(x)="><span></span><span></span></span><br>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{cases}1,当x是有理数的时候(小数部分有限或无限循环的数); \\0,当x是无理数的时候(小数部分无限不循的数)； \\\end{cases}"><span></span><span></span></span>
4.取最值函数:<span class="equation-text" data-index="0" data-equation="y=max\left \{ \ f(x) \right.,\left . \ g(x) \right \}" contenteditable="false"><span></span><span></span></span>,<span class="equation-text" contenteditable="false" data-index="1" data-equation="y=min\left \{ \ f(x) \right.,\left . \ g(x) \right \}"><span></span><span></span></span>
方法：求f(x)和g(x)的交点x0，以x0划分求<span class="equation-text" data-index="0" data-equation="x\leq x_0" contenteditable="false"><span></span><span></span></span>和<span class="equation-text" contenteditable="false" data-index="1" data-equation="x\geq x_0"><span></span><span></span></span>中f(x)和g(x)最大/最小那个函数表示
函数的特性
函数的有界性
有界性(局部概念)
若对于某区间I的所有x都有<span class="equation-text" data-index="0" data-equation="|f(x)|\leq M(常数)成立" contenteditable="false"><span></span><span></span></span>，则称<span class="equation-text" data-index="1" data-equation="f(x)" contenteditable="false"><span></span><span></span></span>在区间I上有界，否则无界。有界<span class="equation-text" contenteditable="false" data-index="2" data-equation="\Rightarrow"><span></span><span></span></span>有上界、下界
<span class="equation-text" contenteditable="false" data-index="0" data-equation="函数f(x)的定义域为D，数集X\subset D"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\exists K_{1}，s.t.f(x)\leq K_{1},\forall x\in X,称f(x)在X上有上界"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\exists K_{2}，s.t.f(x)\geq K_{1},\forall x\in X,称f(x)在X上有下界"><span></span><span></span></span>
常见有界函数：<span class="equation-text" contenteditable="false" data-index="0" data-equation="sinx、cosx以及反三角函数"><span></span><span></span></span>
常用规律：<span class="equation-text" contenteditable="false" data-index="0" data-equation="有界\pm 有界=有界 ，有界\pm无界=无界，有界\cdot有界=有界"><span></span><span></span></span>
函数的单调性
单调性(局部概念)
在某区间上任取<span class="equation-text" contenteditable="false" data-index="0" data-equation="x1<x2"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若f(x1)< f(x2)则说明f(x)在该区间上是单调递增的"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若f(x1)>f(x2)则说明f(x)在该区间上是单调递减的"><span></span><span></span></span>
注：强调单调区间
函数的奇偶性
奇偶性
必定满足x在一个对称区间内，即定义域D关于原点对称，不然不用谈奇偶性
奇函数：<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(-x)=-f(x)"><span></span><span></span></span>，必过(0,0)
偶函数：<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(-x)=f(x)"><span></span><span></span></span>
判断奇偶性的方法
1.定义法：<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(-x)=-f(x)奇函数,f(-x)=f(x)"><span></span><span></span></span>偶函数
2.图像法：奇函数关于原点对称，偶函数关于y轴对称
3.对数专用法：<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(-x)-f(x)=0(偶)，f(-x)+f(x)=0(奇)"><span></span><span></span></span>
4.运算性质：
<span class="equation-text" contenteditable="false" data-index="0" data-equation="奇\pm 奇=奇，偶\pm 偶=偶"><span></span><span></span></span>，同加减同=同
<span class="equation-text" contenteditable="false" data-index="0" data-equation="奇\times 偶=奇,偶\times 偶=偶 ,奇\times 奇=偶"><span></span><span></span></span>，同乘得偶，异乘得奇
<span class="equation-text" contenteditable="false" data-index="0" data-equation="奇(偶)=偶，偶(偶)=偶，偶(奇)=偶，奇(奇)=奇"><span></span><span></span></span>，有偶则偶，全奇则奇
函数的周期性
周期性
在<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)"><span></span><span></span></span>的定义域内对于任意x，恒有<span class="equation-text" contenteditable="false" data-index="1" data-equation="f(x+T)=f(x)"><span></span><span></span></span>则称<span class="equation-text" contenteditable="false" data-index="2" data-equation="f(x)"><span></span><span></span></span>为周期函数(其中T为最小正周期)
常见周期函数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="sinx、cosx以2π为周期"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="tanx、cotx以π为周期"><span></span><span></span></span>
变形后仍为周期函数
1.<span class="equation-text" contenteditable="false" data-index="0" data-equation="sin(Ax+B)"><span></span><span></span></span>
2.<span class="equation-text" contenteditable="false" data-index="0" data-equation="(sinx)^2"><span></span><span></span></span>
3.<span class="equation-text" contenteditable="false" data-index="0" data-equation="|sinx|"><span></span><span></span></span>
两个周期函数变形后周期公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=Asin(\omega x+\phi)+B的周期公式T=\frac{2\pi}{|\omega|}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=Atan(\omega x+\phi)+B的周期公式T=\frac{\pi}{|\omega|}"><span></span><span></span></span>
缩小一半T
1.有绝对值的周期函数，比如:<span class="equation-text" contenteditable="false" data-index="0" data-equation="|sinx|与sinx"><span></span><span></span></span>,周期为T
2.<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=(sinx)^2=\frac{1-cos2x}{2}"><span></span><span></span></span>，周期为T
复合函数
题型一:已知<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=f(x)"><span></span><span></span></span>,求<span class="equation-text" contenteditable="false" data-index="1" data-equation="y=f[\phi(x)]"><span></span><span></span></span>,方法：直接带入即可
分段函数，分析法
分段函数，图像法
<span class="equation-text" contenteditable="false" data-index="0" data-equation="比如：\sqrt{1-x^2}\geq -x"><span></span><span></span></span>
题型二:已知<span class="equation-text" contenteditable="false" data-index="0" data-equation="f[\phi(x)]求f(x)"><span></span><span></span></span>，方法：1.换元法2.凑型法
题型三:复合函数求单调性f[g(x)]，口诀：同增异减
内函数和外函数单调性相同为增函数
内函数和外函数单调性相反为减函数
复合函数可以复合的条件
当且仅当<span class="equation-text" contenteditable="false" data-index="0" data-equation="E^*\neq \varnothing (即D\cap g(E)\neq \varnothing)，函数f(x)与g(x)才能复合"><span></span><span></span></span>
例如：<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=f(u)=arcsinu ,u\in D=[-1,1],u=g(x)=2+x,x\in E=\{x|x\geq 0\}，就不能复合，应为外函数的定义域为D=[-1,1]"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="1" data-equation="与内部函数的值域g(E)=[2,+\propto)不想交"><span></span><span></span></span>
求反函数
反函数是由<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=f(x)"><span></span><span></span></span>确定的<span class="equation-text" contenteditable="false" data-index="1" data-equation="y=(f(x))^{-1}称为反函数"><span></span><span></span></span>
求反函数的步骤
1.先反解出x
解的时候观察式子有ln就用e，有不能直接<span class="equation-text" contenteditable="false" data-index="0" data-equation="x=...的且有根号的使用平方差作两个方程相消"><span></span><span></span></span>
2.再x和y互换
反函数和直接函数之间的关系
1.单调性相同
2.图像关于<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=x"><span></span><span></span></span>对称
3.定义域值域互换
注
自然定义域可以不加定义域
非自然定义域要加
求反函数和复合函数必加定义域
比如结果就是<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=arcsinx可以不加"><span></span><span></span></span>
基本初等函数
1.幂函数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=x^a(a为任意实数)"><span></span><span></span></span>
重点掌握：a=-1、<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{1}{2}"><span></span><span></span></span>、1、2、3
2指数函数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=a^x(a>0,a\neq1)"><span></span><span></span></span>
注
<span class="equation-text" contenteditable="false" data-index="0" data-equation="e^{+\propto}=+\propto"><span></span><span></span></span>,<span class="equation-text" contenteditable="false" data-index="1" data-equation="e^{-\propto}=0"><span></span><span></span></span>
运算法则
3.对数函数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=loga(x)(a>0,a\neq1)"><span></span><span></span></span>
注
消掉ln用e，消掉e用ln
运算法则
<span class="equation-text" contenteditable="false" data-index="0" data-equation="lne^x=x,e^{lnx}=x"><span></span><span></span></span>
4.三角函数与反三角函数
arcsinx与sinx
主值区间：<span class="equation-text" contenteditable="false" data-index="0" data-equation="[-\frac{π}{2},\frac{π}{2}]"><span></span><span></span></span>
arccosx与cosx
主值区间：<span class="equation-text" contenteditable="false" data-index="0" data-equation="[0,π]"><span></span><span></span></span>
arctanx与tanx
主值区间：<span class="equation-text" contenteditable="false" data-index="0" data-equation="[-\frac{π}{2},\frac{π}{2}]"><span></span><span></span></span>
arccotx与cotx
主值区间：<span class="equation-text" contenteditable="false" data-index="0" data-equation="[0,π]"><span></span><span></span></span>
常用公式
倒数关系
<span class="equation-text" contenteditable="false" data-index="0" data-equation="tanx\cdot cotx =1,sinx\cdot cscx=1,cosx\cdot secx=1"><span></span><span></span></span>
商关系
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{sinx}{cosx}=tanx,\frac{cosx}{sinx}=cotx"><span></span><span></span></span>
平方关系
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(sinx)^2+(cosx)^2=1,1+(tanx)^2=(secx)^2,1+(cotx)^2=(cscx)2"><span></span><span></span></span>
诱导公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="sin(-x)=-sinx,cos(-x)=cosx,tan(-x)=-tanx,cot(-x)=-cotx"><span></span><span></span></span>
倍角公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="sin2x=2sinxcosx,cos2x=(cosx)^2-(sinx)^2=2(cosx)^2-1=1-2(sinx)^2"><span></span><span></span></span>,<span class="equation-text" contenteditable="false" data-index="1" data-equation="tan2x=\frac{2tanx}{1-(tanx)^2}"><span></span><span></span></span>
三角降幂公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(sinx)2=\frac{1-cos2x}{2},(cosx)^2=\frac{1+cos2x}{2}"><span></span><span></span></span>
两角和与差的三角函数公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="sinx(x+y)=sinxcosy+cosxsiny"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="sinx(x-y)=sinxcosy-cosxsiny"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="cos(x-y)=cosxcosy+sinxsiny"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="cos(x+y)=cosxcosy-sinxsiny"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="tan(x+y)=\frac{tanx+tany}{1-tanx\cdot tany},tan(x-y)=\frac{tanx-tany}{1+tanx\cdot tany}"><span></span><span></span></span>
万能公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="sinx=\frac{2tan(\frac{x}{2})}{1+(tan(\frac{x}{2}))^2}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="cosx=\frac{1-(tan(\frac{x}{2}))^2}{1+(tan(\frac{x}{2}))^2}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="sinx=\frac{2tan(\frac{x}{2})}{1-(tan(\frac{x}{2}))^2}"><span></span><span></span></span>
和差化积
<span class="equation-text" contenteditable="false" data-index="0" data-equation="sinx+siny=2sin(\frac{x+y}{2})cos(\frac{x-y}{2})"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="sinx-siny=2cos(\frac{x+y}{2})sin(\frac{x-y}{2})"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="cosx+cosy=2cos(\frac{x+y}{2})cos(\frac{x-y}{2})"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="cosx-cosy=-2sinx(\frac{x+y}{2})sin(\frac{x-y}{2})"><span></span><span></span></span>
积化和差
<span class="equation-text" contenteditable="false" data-index="0" data-equation="sinx\cdot cosy=\frac{1}{2}[sin(x+y)+sin(x-y)]"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="cosx\cdot siny=\frac{1}{2}[sin(x+y)-sin(x-y)]"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="cosx\cdot cosx=\frac{1}{2}[cos(x+y)+cos(x-y)]"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="sinx\cdot siny=-\frac{1}{2}[cos(x+y)-cos(x-y)]"><span></span><span></span></span>
注
初等函数（反对幂指三）在其定义域内必连续的，连续必可积
无穷小量比较
设<span class="equation-text" contenteditable="false" data-index="0" data-equation="\alpha,\beta"><span></span><span></span></span>是同一变化过程中的无穷小量，即<span class="equation-text" contenteditable="false" data-index="1" data-equation="lim\alpha=lim\beta=0"><span></span><span></span></span>
方法
相除求极限<span class="equation-text" contenteditable="false" data-index="0" data-equation="lim\frac{\alpha}{\beta}="><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="1" data-equation="\begin{cases}0,\alpha 是\beta的高阶无穷小 \\\propto,\alpha 是\beta的低阶无穷小 \\k,\alpha 是\beta的同阶无穷小\\1,\alpha 是\beta的等价无穷小，记作：\alpha \sim \beta \end{cases}"><span></span><span></span></span>
等价无穷小代换
注
<span class="equation-text" contenteditable="false" data-index="0" data-equation="1.\Box \rightarrow 0，2.乘除法"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="sin\Box \sim \Box,e^\Box -1 \sim \Box"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="tan\Box \sim \Box,ln(1+\Box) \sim \Box"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="arcsin\Box \sim \Box,1-cos\Box \sim \frac{1}{2}\Box ^2"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="arctan\Box \sim \Box,\sqrt{1+\Box} -1 \sim \frac{1}{2}\Box"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(1+\Box)^\alpha -1 \sim \alpha \Box(其中\alpha为常数),\Box - sin\Box \sim \frac{1}{6}\Box^3"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="tan\Box - sin\Box \sim \frac{1}{2}\Box ^3,a^x-1 \sim xlna"><span></span><span></span></span>
函数极限
极限运算法则
如果<span class="equation-text" contenteditable="false" data-index="0" data-equation="limf(x)=A,limg(x)=B"><span></span><span></span></span>那么就有
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{cases}limf(x)\pm limg(x)=A\pm B \\limf(x) \cdot limg(x)=A\cdot B\\lim\frac{f(x)}{g(x)}=\frac{A}{B}(其中B\ne0)\\limkf(x)=kA\end{cases}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="如果lim_{x\to x_{0}} f(x)存在，而n是正整数,则lim_{x\to x_{0}} [f(x)]^n=[lim_{x\to x_{0}}f(x)]^n"><span></span><span></span></span>
抓大头
<span class="equation-text" contenteditable="false" data-index="0" data-equation="注：1.\lim_{x\to \propto},2.多项式相除"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{cases}0,上次小于下次\\\frac{a}{b},上次等于下次 \\\propto,上次大于下次\end{cases}"><span></span><span></span></span>
分子分母同时除以x的最高次项
通分、有理化约去零因子
重要公式
两个重要极限
第一重要极限
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{cases}\lim_{x\to 0} \frac{sinx}{x} = 1 \\\lim_{x\to \propto} xsin\frac{1}{x} =1 \end{cases}"><span></span><span></span></span>
第二重要极限
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{cases}\lim_{x\to 0} (1+x)^{\frac{1}{x}}=e \\\lim_{x\to \propto} (1+\frac{1}{x})^x=e \end{cases}"><span></span><span></span></span>
适用于：<span class="equation-text" contenteditable="false" data-index="0" data-equation="1^\propto 型"><span></span><span></span></span>
其他重要极限
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{n\to \propto} \sqrt[n]{a}(a>0)=1,\lim_{n\to \propto} \sqrt[n]{n}=1"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to +\propto} arctan x=\frac{\pi}{2},\lim_{x\to -\propto} arctanx = -\frac{\pi}{2}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to +\propto}arccotx=0,\lim_{x\to -\propto} arccotx= \pi"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to -\propto} e^x=0,\lim_{x\to +\propto} e^x= \propto"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to 0^+} x^x=1"><span></span><span></span></span>
等价无穷小代换
非零因子乘除可以先算
对数恒等式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x=lne^x"><span></span><span></span></span>
拆
想要拆开算，极限必须存在
((L'Hospital)洛必达法则)洛必达
方法
分子分母各自求导
适用
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{0}{0}"><span></span><span></span></span>或<span class="equation-text" contenteditable="false" data-index="1" data-equation="\frac{\propto}{\propto}"><span></span><span></span></span>
条件
分子分母同趋向于0或无穷大
分子分母在限定的区域内是否分别可导
当两个条件都满足时，再求导并判断求导之后的极限是否存在
若存在，直接得到答案；若不存在，则说明此种未定式无法用洛必达法则解决
如果不确定，即结果仍然为未定式，再在验证的基础上继续使用洛必达法则。
不适用的进行转化
<span class="equation-text" contenteditable="false" data-index="0" data-equation="0\cdot \propto"><span></span><span></span></span>
通过取倒数方式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\propto-\propto"><span></span><span></span></span>
通过根号通过平方差，三角函数通过恒等变形等等
<span class="equation-text" contenteditable="false" data-index="0" data-equation="0^0,1^{\propto},\propto^0"><span></span><span></span></span>
通过公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\rightarrow a} [f(x)]^{g(x)}=\lim_{x\rightarrow a} e^{ln[f(x)]^{g(x)}}=\lim_{x\rightarrow a} e^{g(x)lnf(x)} =e^{lim_{x\rightarrow a }g(x)lnf(x)}"><span></span><span></span></span>
定理
函数极限的唯一性
如果极限<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to x_{0}}存在，那么极限唯一"><span></span><span></span></span>
函数极限的局部有界性(局部)
<span class="equation-text" data-index="0" data-equation="\lim_{x\to x_{0}}f(x)\Leftrightarrow \forall \epsilon>0,\exist \delta>0,使得当0<|x-x_{0}|<\delta时，有|f(x)-A|<\epsilon" contenteditable="false"><span></span><span></span></span>
函数极限的局部保号性(局部)
如果极限<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(x)\rightarrow A(当x\rightarrow x_{0})，而且A>0(或A<0)，那么存在常数\delta>0,使得当0<|x-x_{0}|<\delta时，有f(x)>0(或f(x)<0)"><span></span><span></span></span>
函数极限与数列极限的关系（海涅定理）
可令<span class="equation-text" contenteditable="false" data-index="0" data-equation="n=x"><span></span><span></span></span>
有限个无穷小之和仍为无穷小
有限个无穷小之积仍为无穷小
有界函数(有界变量)与无穷小之积仍为无穷小
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{cases}\lim_{x\to 0} xsin\frac{1}{x}=0 \\\lim_{x\to \propto} \frac{1}{x}sinx=0\end{cases}"><span></span><span></span></span>
不等式性
如果<span class="equation-text" contenteditable="false" data-index="0" data-equation="\phi(x) \geq \psi(x) ,而lim_{x\to x_{0}} \phi(x) =a,lim_{x\to x_{0}} \psi(x)=b,那么a\geq b"><span></span><span></span></span>
夹逼定理
复合函数的求极限运算法则
<span class="equation-text" contenteditable="false" data-index="0" data-equation="lim_{x\to x_{0}} f[g(x)] = lim_{u\to u_{0}} f(u)=A"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若f(x)为一多项式，lim_{x\to x_{0}} f(x) =f(x_{0})"><span></span><span></span></span>
必须是x趋于<span class="equation-text" contenteditable="false" data-index="0" data-equation="x_{0}成立"><span></span><span></span></span>
注
<span class="equation-text" contenteditable="false" data-index="0" data-equation="six\Box:等代sin\Box\sim \Box"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="sin\propto:无穷小乘有界变量"><span></span><span></span></span>
极限不存在常见的两种形式
1.结果<span class="equation-text" contenteditable="false" data-index="0" data-equation="\propto"><span></span><span></span></span>
2.震荡无极限，例如sinx、cosx
极限存在充要条件
左右极限存在且相等
求极限遇到这几种函数必分左右<br>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="arctan\propto,arccot\propto,e^{\propto}(a^{\propto})"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\epsilon-\delta语言精确描述极限(选择题考点)(函数极限局部有界性)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to x_{0}}f(x)\Leftrightarrow \forall \epsilon>0,\exist \delta>0,使得当0<|x-x_{0}|<\delta时，有|f(x)-A|<\epsilon"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to x_{0}^+}f(x)=A\Leftrightarrow \forall \epsilon>0,\exist \delta>0,使得当0<x-x_{0}<\delta时，有|f(x)-A|<\epsilon"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to x_{0}^-}f(x)=A\Leftrightarrow \forall \epsilon>0,\exist \delta>0,使得当-\delta<x-x_{0}<0时，有|f(x)-A|<\epsilon"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to \propto} f(x) =A \Leftrightarrow \forall \epsilon >0,\exists X>0,当|x|>X时，有|f(x)-A|<\epsilon"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to +\propto} f(x) =A \Leftrightarrow \forall \epsilon >0,\exists X>0,当x>X时，有|f(x)-A|<\epsilon"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to -\propto} f(x) =A \Leftrightarrow \forall \epsilon >0,\exists X>0,当x<-X时，有|f(x)-A|<\epsilon"><span></span><span></span></span>
其中<span class="equation-text" contenteditable="false" data-index="0" data-equation="通过\delta=f(x)互相约束\epsilon、\delta"><span></span><span></span></span>线性关系
可以通过<span class="equation-text" contenteditable="false" data-index="0" data-equation="0<|x-x_{0}|<\delta可以知道"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="0<x-x_{0}<\delta \Rightarrow x_{0}<x<x_{0}+\delta"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="-\delta<x-x_{0}<0 \Rightarrow x_{0}-\delta<x<x_{0}"><span></span><span></span></span>
可以通过<span class="equation-text" contenteditable="false" data-index="0" data-equation="|f(x)-A|<\delta 可以知道"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="-\epsilon< f(x)-A<\delta,A-\epsilon<f(x)<A+\epsilon"><span></span><span></span></span>
数列极限
数列极限求解
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{n\to \propto} a^n 求f(x)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{cases}0,|a|<1 \\\propto,|a|>1 \\1,a=1\\不存在,a=-1\end{cases}"><span></span><span></span></span>
数列极限精确定义<span class="equation-text" contenteditable="false" data-index="0" data-equation="\epsilon-N定义"><span></span><span></span></span>
对于任意给定的正数<span class="equation-text" contenteditable="false" data-index="0" data-equation="\epsilon(不论有多小)，总是存在一个正数N，使得对于n>N时的一切x_{n}，都有|x_{n}-a|<\epsilon都成立，称常数a是数列{a_{n}}的极限"><span></span><span></span></span>,\<span class="equation-text" contenteditable="false" data-index="1" data-equation="\lim_{n\to \propto} x_{n} =a"><span></span><span></span></span>或<span class="equation-text" contenteditable="false" data-index="2" data-equation="x_n{} \rightarrow a (n\rightarrow \propto)"><span></span><span></span></span>
定理
极限的唯一性
<span class="equation-text" contenteditable="false" data-index="0" data-equation="如果数列 \{x_{n}\}收敛，则它的极限唯一"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="若\lim_{n\to \propto} a_{n} =a,则lim_{n\to \propto} |a_{n}| = |a|"><span></span><span></span></span>
数列的有界性
<span class="equation-text" contenteditable="false" data-index="0" data-equation="对于数列\{x_{n}\},如果存在正数M，使得对于一切x_{n}都满足不等式|x_{n}|\leq M则称数列\{x_{n}\}有界，否则无界"><span></span><span></span></span>
收敛数列的保号性
<span class="equation-text" contenteditable="false" data-index="0" data-equation="如果数列\{x_{n}\}收敛于a,且a>0(或a<0)，那么存在正整数N，当n>N时，有x_{n}>0(或x_{n}<0)"><span></span><span></span></span>
保不等式性
<span class="equation-text" contenteditable="false" data-index="0" data-equation="设\{a_{n}\}与\{b_{n}\}"><span></span><span></span></span>均是收敛数列，存在正整数N，使得当n>N时，有<span class="equation-text" contenteditable="false" data-index="1" data-equation="a_{n} \leq b_{n} ,则lim_{n\to \propto} a_{n}\leq lim_{n\to \propto} b_{n}"><span></span><span></span></span>
迫敛性(夹逼定理)
如果数列<span class="equation-text" contenteditable="false" data-index="0" data-equation="X_{n}、Y_{n}、Z_{n}"><span></span><span></span></span>满足
<span class="equation-text" contenteditable="false" data-index="0" data-equation="X_{n}\leq Y_{n}\leq Z_{n}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{n\to \propto} Y_{n} =a,\lim_{n\to \propto} Z_{n} =a"><span></span><span></span></span>
那么数列<span class="equation-text" contenteditable="false" data-index="0" data-equation="X_{n}"><span></span><span></span></span>的极限存在，且<span class="equation-text" contenteditable="false" data-index="1" data-equation="\lim_{n\to \propto} X_{n}=a "><span></span><span></span></span>
数列求极限工具
四则运算法则
若<span class="equation-text" contenteditable="false" data-index="0" data-equation="\{a_{n}\}"><span></span><span></span></span>与<span class="equation-text" contenteditable="false" data-index="1" data-equation="\{b_{n}\}"><span></span><span></span></span>均是收敛数列，则<span class="equation-text" contenteditable="false" data-index="2" data-equation="\{a_{n}\pm b_{n}\},\{a_{n}\cdot b_{n}\}也是收敛数列，且有"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{n\to \propto} (a_{n} \pm b_{n})=\lim_{n\to \propto} a_{n} \pm \lim_{n\to \propto} b_{n}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{n\to \propto} (a_{n} \cdot b_{n})=\lim_{n\to \propto} a_{n} \cdot \lim_{n\to \propto} b_{n}"><span></span><span></span></span>
单调有界定理
单调有界数列必有极限（单调递增有上界必有极限；单调递减有下界必有极限）
子列
若<span class="equation-text" contenteditable="false" data-index="0" data-equation="\{a_{n}\}为数列,\{n^k\}为正整数集N_{+}的无限集，且n_{1}<n_{2}<...n_{k}<..."><span></span><span></span></span>，则数列称为<span class="equation-text" contenteditable="false" data-index="1" data-equation="\{a_{n}\}的一个子列，记作\{a_{n_{k}}\}"><span></span><span></span></span>
数列敛散性判断工具
抓大头
注
数列极限不能直接使用洛必达法则，必须使用海涅定理
海涅定理（一般<span class="equation-text" contenteditable="false" data-index="0" data-equation="n\rightarrow \propto,大部分令\frac{1}{n} = x，较少令n=x"><span></span><span></span></span>）
<span class="equation-text" contenteditable="false" data-index="0" data-equation="lim_{x\to x_{0}} f(x)= A \Leftrightarrow \forall x_{n} \rightarrow x_{0} ,lim_{n\to \propto} f(x_{n}) =A"><span></span><span></span></span>
例题
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{n\to \propto} (\frac{1}{n})^{sin\frac{1}{n}}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to 0^+}(x)^{sinx}=1,由海涅定理得\lim_{n\to \propto} (\frac{1}{n})^{sin\frac{1}{n}}=1"><span></span><span></span></span>
数列极限和函数极限
区别
数列极限
<span class="equation-text" contenteditable="false" data-index="0" data-equation="n\rightarrow \propto"><span></span><span></span></span>
函数极限
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x\rightarrow \propto"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x\rightarrow x_{0}"><span></span><span></span></span>
做极限题目思路
是否分必分左右极限
必分
分左右极限算再合并
不分
1.常/为定式
未定式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{0}{0}、\frac{\propto}{\propto}洛必达"><span></span><span></span></span>
2.化简
约分(约公因子)<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{\propto}{\propto}"><span></span><span></span></span>
通分(分式相减)<span class="equation-text" contenteditable="false" data-index="0" data-equation="\propto-\propto"><span></span><span></span></span>
有理化(根式相减)<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{0}{0}"><span></span><span></span></span>
此处的根式可以考虑使用拉格朗日中值定理和夹逼定理解决
拉格朗日+夹逼定理
拉格朗日+判定等价
<span class="equation-text" contenteditable="false" data-index="0" data-equation="0\cdot \propto转化成"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\frac{0}{0}"><span></span><span></span></span>、<span class="equation-text" contenteditable="false" data-index="1" data-equation="\frac{\propto}{\propto}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="0^0通过取e"><span></span><span></span></span>转化
<span class="equation-text" contenteditable="false" data-index="0" data-equation="0\cdot \propto"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(\propto)^0通过取e转化"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="0\cdot \propto"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="1^\propto通过利用第二重要极限"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to 0} (1+x)^{\frac{1}{x}}=e"><span></span><span></span></span>
特别注意
<span class="equation-text" contenteditable="false" data-index="0" data-equation="lim \frac{C}{0}"><span></span><span></span></span><span class="equation-text" contenteditable="false" data-index="1" data-equation="=\begin{cases}+\propto,C>0\\-\propto,C<0 \end{cases}"><span></span><span></span></span>
渐近线
垂直渐近线(垂直x轴，一般怀疑分母为0的点)
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to x_{0}^+} f(x)=\propto或\lim_{x\to x_{0}^-} f(x)=\propto"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x=x_{0}为一条垂直渐近线"><span></span><span></span></span>
水平渐近线(平行于x轴的渐近线)
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to +\propto} f(x)= b或\lim_{x\to -\propto} f(x)= b"><span></span><span></span></span>
y=b为一条水平渐近线
注：水平与斜在相同过程下不共存
斜渐近线(一般渐近线)
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to \propto} \frac{f(x)}{x}=a"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to \propto} [f(x)-ax]=b"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="y=ax+b为一条斜渐近线"><span></span><span></span></span>
注：水平与斜在相同过程下不共存
函数连续性判断
考点
1.区间连续
考定义域
给一个函数在起定义域上连续求其中含着的a、b的值
2.点连续
考分段函数
给一个函数让你证明它在区间内是连续的
连续条件
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x\to x_{0}}f(x)=f(x_{0}) \Leftrightarrow "><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{cases}\lim_{x\to x_{0}^-}f(x)=f(x_{0}) \\\lim_{x\to x_{0}^+}f(x)=f(x_{0}) \\f(x)在x_{0}处有定义\end{cases}"><span></span><span></span></span>
连续原始定义
<span class="equation-text" contenteditable="false" data-index="0" data-equation="1.\lim_{\nabla x\to0} \nabla y =0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="2.\lim_{\nabla x \to 0}f(x_{0}+\nabla x)-f(x_{0})=0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="3.\lim_{x\to x_{0}} f(x)=f(x_{0})"><span></span><span></span></span>
间断点判断
判定间断点的步骤
1.求定义域
2.确定定义域内x不等于谁，谁就是间断点
3.根据左右间断点存在的情况判定是第几类间断点
间断点类型
第一类间断点（左右极限都存在）
可去间断点（左右极限相等不等于该点处的函数值）
跳跃间断点（左右极限不相等）
第二类间断点（左右间断点至少有一个不存在）
无穷间断点（左右极限至少有一个不存在）
振荡间断点（震荡无极限）
几大定理
最值定理
条件
f(x)在闭区间[a,b]上连续
结论
f(x)在[a,b]上一定有最大值M和最小值m
介值定理
条件
f(x)在闭区间[a,b]上连续
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(a)\neq f(b)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(a)<u<f(b)或f(b)<u<f(a)"><span></span><span></span></span>
结论
至少存在一点<span class="equation-text" contenteditable="false" data-index="0" data-equation="\xi \in(a,b) "><span></span><span></span></span>,使得<span class="equation-text" contenteditable="false" data-index="1" data-equation="f(\xi)=u"><span></span><span></span></span>
注
这里的结论可以放缩到<span class="equation-text" contenteditable="false" data-index="0" data-equation="\xi \in[a,b]"><span></span><span></span></span>区间上
零点定理
条件
f(x)在闭区间[a,b]上连续
<span class="equation-text" contenteditable="false" data-index="0" data-equation="f(a)\cdot f(b)<0"><span></span><span></span></span>
结论
至少存在一点<span class="equation-text" data-index="0" data-equation="\xi \in(a,b) " contenteditable="false"><span></span><span></span></span>,使得<span class="equation-text" data-index="1" data-equation="f(\xi)=0" contenteditable="false"><span></span><span></span></span>
方程根的存在性判断
1.由零点定理，端点值异号至少一个根
2.仅一个根证明：在1.的基础上判定函数单调性
3.k个根证明：将区间分为k个区间，再重复2.的步骤
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